All Questions
36
questions
2
votes
1
answer
92
views
About the spectrum family of the multiplication operator
Let $<,>$ be the inner product of $L^2(\mathbb{R})$.
For a measurable function $F:\mathbb{R}^d\to\mathbb{R}$, we define
a multiplication operator $M_F:L^2(\mathbb{R}^d)\to L^2(\mathbb{R}^d)$, ...
- 61
2
votes
1
answer
202
views
Spectrum of a sum of self-adjoint operators
This is a "sequel" to that question where I explain why I need the spectrum of an operator given as the sum of a convolution and a function multiplication. Here, I am considering the ...
- 1,139
2
votes
2
answers
266
views
Compactness of subset of trace-class operators on a Hilbert space
Consider an infinite-dimensional, complex and separable Hilbert space $H$ and let $\mathcal I(H)$ denote the space of trace-class operators.
The set of density operators is defined by $$\mathcal S(H):...
- 428
1
vote
1
answer
84
views
Operator theory: nonnegative operator $Q$ less than an orthogonal projection $P$ satisfies $PQ=QP$
Suppose that we have a Hilbert space $\mathcal{H}$, an orthogonal projection $P$ (i.e. $P=|\Psi\rangle\langle\Psi|$ for some $\Psi\in\mathcal{H}$ with $\|\Psi\|=1$) and another non-negative bounded ...
- 85
1
vote
1
answer
65
views
An inequality for Schatten norms with a compact self-adjoint operator
Let $Q$ be a compact self-adjoint operator on $L^2(\mathbb R,\mathbb C)$. Notice that $(1-\Delta)^{s}$ is a positive (and hence self-adjoint) operator on $H$ for any $s>0$.
We denote by $\mathfrak ...
- 461
0
votes
0
answers
63
views
What is the most general Self-Adjoint operator acting on $L^{2}(\mathbb{R})$?
What is the most general Self-Adjoint operator acting on $L^{2}\mathbb{R}$?
My hypothesis is that the aswer to my question will be that the most general Self-Adjoint operator $A$ acting on $L^{2}(\...
- 71
0
votes
0
answers
27
views
Question regarding conjugate operators and the harmonic operator.
Let us consider the operator $\hat{n}=\hat{a}^\dagger\hat{a}$ as the number operator of the harmonic oscillator. Let $|n\rangle$ be the eigenstates. Then we can say : $$\hat{n}|n\rangle=n|n\rangle$$
I'...
1
vote
1
answer
67
views
Equivalence between two definitions of hermitian adjoint
Given the two definitions of hermitian adjoint:
$(1): <\Psi|A|\Phi>=(<\Psi|A^+|\Phi>)^*$
$(2): <\Psi|A\Phi>=<A^+\Psi|\Phi>$
I want to show that they are equivalent
However I ...
- 541
1
vote
1
answer
74
views
Action of an operator-valued function of the momentum operator $\hat{p}$ and its unboundedness
I am currently dealing with an operator-valued function
$f(\hat{T})$ of the following kind:
$$f(\hat{T}) =\sqrt{1 + b\hat{T}^2} $$
where $b$ $\in \mathbb{R}$ and $\hat{T}$ is a linear operator acting ...
- 31
2
votes
1
answer
105
views
Consequences of Deficiency indices theorem (Von Neumann theory)
Let $T: \operatorname{dom}(T) \rightarrow \scr H$ be a symmetric operator.
$T$ admits self-adjoint extensions $\iff$ $d_+ = d_-$, where $d_\pm = \dim \ker(T^\dagger \pm i \mathbb{I})$
If $d_+ = d_-$,...
- 141
1
vote
1
answer
106
views
How to prove that frame functions on Hilbert spaces are additive?
Let $\newcommand{\calH}{\mathcal{H}}\newcommand{\eff}{\operatorname{Eff}(\calH)}\calH$ be some separable Hilbert space, and denote with $\eff$ the set of effects on $\calH$, that is, the set of ...
- 7,095
0
votes
1
answer
142
views
Prove convergence of series under trace-norm topology
Let $H$ be a separable Hilbert space, $T(H)$ the set of trace-class operators, and $D(H)\subset T(H)$ the set of density operators (i.e., positive and having trace 1).
For any unit vector $\vert \...
- 1,254
1
vote
0
answers
26
views
Strong convergence of unitary propagators for the time-dependent Schrödinger problem
Consider a family of self-adjoint operators $H_n(\alpha)$ on a Hilbert space $\mathcal{H}$, with $n\in\mathbb{N}$ and $\alpha$ ranging in some bounded real interval $I$, such that, for every $\alpha\...
- 391
2
votes
1
answer
97
views
$\langle u| \langle v| A |u\rangle |v\rangle \geq 0$ for any $|u\rangle \in \mathcal{H}_1$ and $|v\rangle \in \mathcal{H}_2$. Then $A$ is Hermitian?
$\mathcal{H}_1$ and $\mathcal{H}_2$ are complex Hilbert spaces. $A$ is an operator from $\mathcal{H}_1\otimes \mathcal{H}_2$ to $\mathcal{H}_1\otimes \mathcal{H}_2$.
Suppose that $\langle u| \langle ...
- 861
4
votes
1
answer
903
views
Stone's theorem and the spectral theorem
I am struggling to formally derive the expression found in the Stone's theorem for one-parameter unitary groups. I am aware that this can be done by using the spectral theorem. I am mostly interested ...
- 43